We observed a large sunspot in NOAA Active Region 9535 on 2001 July 12, 15, 17, and 19 with three instruments (CDS, EIT, and MDI) on board SOHO. EUV emission lines of O IV (formed at logarithmic temperature of 5.2), O V (5.4), Ne IV (5.2), Ne V (5.5), Ne VI (5.6), Ne VII (5.7), and Ca X (5.9) revealed a large bright plume within the sunspot penumbra on July 12 and 15, a smaller, dimmer plume on July 17, and no plume on July 19. Downflows of 25 km s-1 or more were measured within the sunspot plume on July 12 and 15. By July 17 the downflow area had shrunk in size, the downflows had diminished in magnitude, and upflows were measured in the umbra and parts of the penumbra outside the plume. By July 19 downflows were no longer observed, but had been replaced entirely with upflows of 15-25 km s-1 in the umbra and portions of the penumbra. This is the first time that the disappearance of a sunspot plume has been observed to occur simultaneously with a dramatic change in flow velocity pattern in the sunspot plume and umbra. On July 12 upflows northeast of the sunspot were observed along with the downflows in the plume. Electron density measurements based on intensity ratios of the O IV lines at 625.8 and 554.5 Å indicate a significantly greater value in the upflow zone than in the plume, consistent with siphon flow as the driver of the observed velocities; however, the line at 625.8 Å is very weak and blended in the red wing of the Mg X line at 624.9 Å, so derived densities are highly uncertain.